Maps that must be affine or conjugate affine: a problem of Vladimir Arnold

Arnold Mathematical Journal. 2020. No. 6. P. 213–229.

Gorinov A., Howard R., Johnson V., McNulty G.

A $k$-flat in a vector space is a $k$\hl{-}dimensional affine subspace. Our basic
result is that an injection $T\cn \C^n\to \C^n$\hl{ }that for some   
$k\in\{1,2,\cd n-1\}$ $T$ maps all $k$-flats to flats of $\C^n$ and is either
continuous at a point or Lebesgue measurable, is either an affine map or a  
conjugate affine map. An analogous result is proven for injections of the
complex projective spaces. In the case of continuity at a point this is
generalized in several directions, the main one being that the complex numbers
can \hl{be} replaced by \hl{a finite-dimensional division algebra over} an Archimedean ordered field. \hl{We also prove injective versions of the Fundamental Theorems of affine and projective geometry and give a counter-example to the surjective version of the latter.}
This extends work of \hl{A.~G.~}Gorinov on a problem of V.~I.~Arnold.

Keywords: semiaffine maps, semilinear maps, Archimedian fields, полуаффинные отображения, полулинейные отображения, архимедовы поля